# 3 Systems of Equations Calculator + Online Solver With Free Steps

The 3 systems of equations calculator is used to solve equations for the three variables $x$, $y$, and $z$.

The three systems of equations are a set of three equations with three variables.  It takes three equations as input, rearranges the equations, and solves for the values of $x$, $y$, and $z$.

This calculator can also solve second and third-degree higher-degree equations, giving complex solutions for $x$, $y$, and $z$. If the system of equations is linear, the calculator gives three real solutions.

## What Is a 3-Systems of Equations Calculator?

The 3-systems of equation calculator is an online calculator that solves three equations with three distinct variables using different methods and gives the solution for the unknown variables.

The different methods used to solve the equations are the substitution method, the elimination method, and the graphing method. The calculator only uses the first two methods for solving the system.

## How To Use the 3 Systems of Equations Calculator?

You can use the 3 systems of equations calculator by entering the three equations and pressing the submit button.

Following is a detailed explanation of the steps that are required to use the 3 systems of equations calculator.

### Step 1

Enter the three equations in the blocks titled Eqn 1, Eqn 2, and Eqn 3, respectively. The three variables used by default are $x$, $y$, and $z$ but the user can also use different variables. The equations by default are linear but the user can also find solutions for higher-order equations.

### Step 2

Enter the Submit button for the calculator to process the three input equations.

### Output

The output window shows the following blocks:

### Input

The input window shows the interpreted input of the calculator. From here, the user can check whether the entered equations are correct or incorrect. If the input is incorrect, the window displays “Not a valid input, please try again.”

### Alternate Forms

This window shows some of the alternate forms of the three equations by rearranging them for different variables on one side.

### Solutions

This window shows the obtained solutions from the three systems of equations. The solutions are the values of unknown variables in the equations.

The user can also click on “Need a step-by-step solution for this problem?” to view all the steps for the particular system of equations.

## Solved Examples

Following are some solved examples of the 3 Systems of equations calculator.

### Example 1

For the three system of equations:

$2x + y + z = 7$

$2x – y + 2z = 6$

$x – 2y + z = 0$

Find the values of $x$, $y$, and $z$.

### Solution

First, enter the three equations in the calculator’s input window. Press “Submit” for the calculator to show results.

The calculator shows the input equations typed by the user, then it displays the solutions for $x$, $y$, and $z$ as follows:

$x = 1$

$y = 2$

$z = 3$

The calculator also gives the alternate forms of the three equations by rearranging them for the third variable z.

For equation 1:

$2x + y + z = 7$

$z = – 2x – y + 7$

For equation 2:

$2x – y + 2z = 6$

$2x + 2z = 6 + y$

Taking 2 as common from left-hand side:

$2 ( x + z ) = y + 6$

Dividing by 2 on both sides gives us:

$x + z = \frac{y}{2} + 3$

So:

$z = – x + \frac{y}{2} + 3$

For equation 3:

$x – 2y + z = 0$

Adding 2y on both sides gives us:

$x + z = 2y$

So the final value is:

$z = 2y – x$

### Example 2

For the three system of equations:

$3x – 2y + 4z = 35$

$-4x + y – 5z = -36$

$5x – 3y + 3z = 31$

Solve for $x$, $y$, and $z$.

### Solution

Enter the three equations in the input window and press “Submit” for the calculator to show its results, which are as follows:

First, the calculator shows the interpreted input equations.

Then it solves for the values of $x$, $y$, and $z$, which are:

$x = -1$

$y = -5$

$z = 7$

The next window shows the alternate forms of the three input equations.

For equation 1:

$3x – 2y + 4z = 35$

Rearranging equation 1:

$3x + 4z = 2y + 35$

This is the first alternate form shown on the calculator.

Now, dividing by 4 on both sides:

$\frac{3x}{4} + z = \frac{y}{2} + \frac{35}{4}$

So the equation becomes:

$z = \frac{-3x}{4} + \frac{y}{2} + \frac{35}{4}$

This is the second alternate form.

For equation 2:

$-4x + y – 5z = -36$

Multiplying by -1 gives:

$4x – y + 5z = 36$

Rearranging equation 2:

$4x + 5z = y + 36$

This is the first alternate form shown on the calculator.

Dividing by 5 on both sides:

$\frac{4x}{5} + z = \frac{y}{5} + \frac{36}{5}$

So:

$z = \frac{-4x}{5} + \frac{y}{5} + \frac{36}{5}$

For equation 3:

$5x – 3y + 3z = 31$

$5x + 3z = 3y + 31$

This is the first alternate form shown on the calculator.

Rearranging the equation:

$3z = -5x + 3y + 31$

Dividing by 3 on both sides gives us:

$z = \frac{-5x}{3} + y + \frac{31}{3}$

The above equation is another alternate form.